CAT 2020 Slot 2 QA Question 26: In how many ways can a pair of integers $(x, a)$ be chosen such that $x^2 - 2|x| + |a - 2| = 0$?

Question

In how many ways can a pair of integers $(x, a)$ be chosen such that $x^2 - 2|x| + |a - 2| = 0$?

Accepted Answer

Let
$t = |x|$
where $t$ is a non-negative integer.

The equation becomes
$t^2 - 2t + |a - 2| = 0$
$|a - 2| = 2t - t^2$

The right side must be non-negative.
$2t - t^2 \geq 0$
$t(2 - t) \geq 0$
So $t$ can be $0, 1$ or $2$.

If $t = 0$, then $x = 0$ and $|a - 2| = 0$. This gives $1$ pair.
If $t = 1$,

In how many ways can a pair of integers (x,a)(x, a)(x,a) be chosen such that x2−2∣x∣+∣a−2∣=0x^2 - 2|x| + |a - 2| = 0x2−2∣x∣+∣a−2∣=0?